A nonstandard proof of a lemma from constructive measure theory

The conceptual gulf between constructive analysis (in the sense of Bishop [1]) and nonstandard analysis (in the sense of Robinson [7]) is vast, despite the historical genesis of both methodologies from foundational considerations. Those who use nonstandard arguments often say of their proofs that they are “constructive modulo an ultrafilter”; implicit in this statement is the suggestion that such arguments might give rise to genuine constructions. On the other hand, to some constructivists nonstandard analysis represents the worst extreme of nonconstructive (i.e., classical) mathematics; see, e.g., Bishop’s review of Keisler’s calculus text [2]). This note gives two nonstandard proofs of an existence result (Theorem 2.1 below) which is central to the constructive treatment of integration, and for which the original proof (Section 7 in [3]) was very difficult. Even the shorter proof (in [5]) is relatively difficult, and the argument that the algorithm provided there actually works is far from transparent. By contrast, these two proofs are very short and straightforward. Of course, much of the difficulty of the proofs in [3, 5] is due to their constructive nature. The nonstandard proofs in Section 3 below are not at all constructive. However, they do give insight both into why the lemma is true, and into why the constructive proofs are so difficult. Moreover, even the constructive approach can be simplified using a bit of nonstandard reasoning; see Section 5. Another justification for proofs of this type is the recent work by Eric Palmgren [6], who demonstrates that many nonstandard arguments can in fact be interpreted as constructive. While his methods do not immediately apply to the proofs here, there is some hope that